What is Exponential: Definition and 1000 Discussions
In mathematics, the exponential function is the function
f
(
x
)
=
e
x
,
{\displaystyle f(x)=e^{x},}
where e = 2.71828... is Euler's constant.
More generally, an exponential function is a function of the form
f
(
x
)
=
a
b
x
,
{\displaystyle f(x)=ab^{x},}
where b is a positive real number, and the argument x occurs as an exponent. For real numbers c and d, a function of the form
f
(
x
)
=
a
b
c
x
+
d
{\displaystyle f(x)=ab^{cx+d}}
is also an exponential function, since it can be rewritten as
a
b
c
x
+
d
=
(
a
b
d
)
(
b
c
)
x
.
{\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}
The exponential function
f
(
x
)
=
e
x
{\displaystyle f(x)=e^{x}}
is sometimes called the natural exponential function for distinguishing it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since
a
b
x
=
a
e
x
ln
b
{\displaystyle ab^{x}=ae^{x\ln b}}
As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:
d
d
x
b
x
=
b
x
log
e
b
.
{\displaystyle {\frac {d}{dx}}b^{x}=b^{x}\log _{e}b.}
For b > 1, the function
b
x
{\displaystyle b^{x}}
is increasing (as depicted for b = e and b = 2), because
log
e
b
>
0
{\displaystyle \log _{e}b>0}
makes the derivative always positive; while for b < 1, the function is decreasing (as depicted for b = 1/2); and for b = 1 the function is constant.
The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function is its own derivative:
This function, also denoted as exp x, is called the "natural exponential function", or simply "the exponential function". Since any exponential function can be written in terms of the natural exponential as
b
x
=
e
x
log
e
b
{\displaystyle b^{x}=e^{x\log _{e}b}}
, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by
The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is a complicated expression. The graph of
y
=
e
x
{\displaystyle y=e^{x}}
is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation
d
d
x
e
x
=
e
x
{\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}}
means that the slope of the tangent to the graph at each point is equal to its y-coordinate at that point. Its inverse function is the natural logarithm, denoted
log
,
{\displaystyle \log ,}
ln
,
{\displaystyle \ln ,}
or
log
e
;
{\displaystyle \log _{e};}
because of this, some old texts refer to the exponential function as the antilogarithm.
The exponential function satisfies the fundamental multiplicative identity (which can be extended to complex-valued exponents as well):
It can be shown that every continuous, nonzero solution of the functional equation
f
(
x
+
y
)
=
f
(
x
)
f
(
y
)
{\displaystyle f(x+y)=f(x)f(y)}
is an exponential function,
f
:
R
→
R
,
x
↦
b
x
,
{\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x},}
with
b
≠
0.
{\displaystyle b\neq 0.}
The multiplicative identity, along with the definition
e
=
e
1
{\displaystyle e=e^{1}}
, shows that
e
n
=
e
×
⋯
×
e
⏟
n
factors
{\displaystyle e^{n}=\underbrace {e\times \cdots \times e} _{n{\text{ factors}}}}
for positive integers n, relating the exponential function to the elementary notion of exponentiation.
The argument of the exponential function can be any real or complex number, or even an entirely different kind of mathematical object (e.g., matrix).
The ubiquitous occurrence of the exponential function in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.
Homework Statement
I'm writing a lab report and I'm having trouble understanding how to set up these experimental equations.
I understand Linear (y=mx+b) but my professor wants us to use separate equations to illustrate power and exponential forms.
Power : y=bx^m
Exponential: y=be^(mx)...
Homework Statement
Use the function y=2e^-0.5x^2 to answer the following questions
a) state the domain
b) Determine the intercepts, if any
c) Discuss the symmetry of the graph
d) Find any asymptotes
e)determine the intervals of increase and decrease
f)what is the maxima and/or minima...
5(0.10)^x = 4(0.12)^x... x = ? getting aggravated
Homework Statement
5(0.10)^x = 4(0.12)^x
this problem is driving me nuts. i graphed it and got x~1.2239011 but I can't find it on paper.
i have been using the property ln(a^x) = x*ln(a)
The Attempt at a Solution
i tried doing...
For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value
from searching in internet I found that
E{X|X>a}=a+E{x}
but I can not prove it
Help please
Homework Statement
For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value
Homework Equations
The Attempt at a Solution
Fix some constant 0<\alpha \leq 1, and denote the floor function by x\mapsto [x]. The conjecture is that there exists a constant \beta > 1 such that
\beta^{-n} \sum_{k=0}^{[\alpha\cdot n]} \binom{n}{k} \underset{n\to\infty}{\nrightarrow} 0
Consider this conjecture as a challenge. I don't...
Homework Statement
A manufacturing process produces 92% good chips (G) and 8% bad chips (B).
The lifetime, in seconds, of chips is exponentially distributed E(\lambda).
For good chips, \lambda1=1/20000 For bad chips,\lambda2=1/1000
Every chip is tested for 50 seconds prior to leaving the...
Let's say you have a random sample of 5 values that are drawn from an exponential distribution with a mean of 8.
How do I find the distribution of Ybar, which is the sample mean of the 5 random variables? [Note: Ybar = 1/5(Y₁+Y₂+Y₃+Y₄+Y₅)]
I know that for an exponential distribution with...
Please help i have these two questions and i am Stuck!
Question 1
The linear equation you have found (y=.771x+1.609) is in the form of y= mx +c. it should be however more appropreiatly be considered as being in the form, lny = mx + lnA. by using appropriat logarithmic and exponential laws...
Hi all,
I am now doing revision for one of the statistics module.
I am having some difficulty to proove the following:
Given n iid Exponential distribution with rate parameter \mu,
using convolution to show that the sum of them is Erlang distribution with density
f(x) = \mu...
Modeling with exponential and logarithmic functions help?
Homework Statement
Use Newton's Lay of Cooling, T = C + (T0 - C)e-kt, to solve this exercise. At 9:00 A.M., a coroner arrived at the home of a person who had died during the night. The temperature of the room was 70 degrees F, and at...
I'm reading that if you have a complex exponential exp(iω0n) where n is in the set of integers, then unlike for the case of a continuous independent variable, the set of complex exponentials that is harmonically-related to this one is finite. I.e. there is only a finite number of distinct...
I got a question here, and I am stuck at understanding the step of the solution. Any help will be appreciated.
http://img841.imageshack.us/img841/6589/40155869.jpg
I would like to know how to get from the second to the third step, where the summation comes in.
It looks like multiplication of...
I am trying to understand the physics of an exponential horn. I got most of my references online and according to the Horn Physics by Martin King, the volume velocity at the mouth is greater than the applied volume velocity at the throat (the horn is driven by a speaker/buzzer).
This is what i...
Hi everyone,
There's something that's kind of been bugging me about applying exponential decay formulas to real world phenomena. For example let's say the discharging of a parallel plate capacitor. Let's consider the negative plate. As it discharges excess electrons leave the plate. The...
Homework Statement
im wondering if e^i(x+y) + e^-i(x+y)
can be simplified to 2cosxcosy
Homework Equations
2cosx= e^ix + e^-ix
so i separated e^i(x+y) + e^-i(x+y)
into;
e^ix(e^iy) + e^-ix(e^-iy)
can that become 2cosxcosy?
Homework Statement
Determine the limit and then prove your claim.
limx\rightarrow\infty (1+\frac{1}{x^2} }) xHomework Equations
I know that the formal definition that I need to use to prove the limit is:
{limx\rightarrow\infty (1+\frac{1}{x^2})x=1}={\forall \epsilon>0, \exists N > 0, \ni x>N...
Hi, I'm in a signal processing class, and I'm having some trouble with complex numbers. As an example, I've attached a pretty simple question about an exponential Fourier series.
I don't find these questions particularly hard, it just takes me ridiculously long to do them. I mean, after I...
Homework Statement
Find the exact value of P(|Y- μ| < 1.4σ) for an exponential random variable with parameter β.
Homework Equations
The only equation that I can think of is the exponential distribution equation:
∫(1/β)e^(-y/β)
The Attempt at a Solution
I have been unable to...
Hi,
I have a conceptual question. Looking at exponential smoothing methods I came across relationship between the autocorrelation function and lambda. It says that if the time series doesn't apper to be autocorrelated then lambda is expected to have a low value :confused: .Any help will be...
Homework Statement
Consider a Poisson process for which events occur at a rate of 2 per hour.
(a) Give the probability that the time until the first event occurs exceeds 2 hours. Use an exponential distribution to find the probability.Homework Equations
The Attempt at a Solution
\lambda = 1/2...
1.\frac{dx}{dt}= \stackrel{9 -12}{2 -1}
x(0)=\stackrel{-13}{-5}
So I seem to be having issues with this problem
There are 2 eigenvalues that I obtained from setting
Det[A-rI]=0
That gave me r^{2}-8r+15=0
solving for r and finding the roots i got
(r-3)*(r-5)=0
so the...
Homework Statement
I need to use the relation exp(i*Pi/4) = (1+i)/sqrt2 but I'd like to know where it came from. I am clueless about how to arrive at this.
Hello,
Suppose you observe some foam. The foam is formed by a set of bubbles, and each bubble blows up after a random time. The density function of the time each bubble will take to blow up is probably exponential, with rate lambda. The total amount of foam (Q) must also decay exponentially...
Homework Statement
At how many points in the xy-plane do the graphs of y=x^{12} and y=2^{x} intersect?
Homework Equations
none
The Attempt at a Solution
I have no idea what to do. I thought of trying to narrow it down to some intervals where the graphs may cross, but, since they're...
Homework Statement
Solve
y''(t) - k^2 y(t) = e^{-a|t|} where a and k are both positive and real.
Homework Equations
The solution was obtained trough a Fourier transform.
The Attempt at a Solution
I got the solution
y(t) = \frac{ke^{-at} - ae^{-kt}}{k(a^2 - k^2)}
but when i...
D_n=\frac{1}{\pi}\int_0^\pi\sin{(t)}\cdot e^{-i2nt}dt=\frac{2}{\pi(1-4n^2)}
I have no idea on how they get from one side of the equation symbol to the other, can i get some tips and tricks ?
I have try ed writing sint as an exp function, but i don't feel it gets me anywhere close.
Homework Statement
I am searching for a proof of convergence of matrix exponential series. Where I can find it?
Thanks!
Homework Equations
The Attempt at a Solution
Homework Statement
Find the number of ways to place 8 toys amongst 4 children where 1 child gets at least two toys.
Homework Equations
(x^2/2! + x^3/3! + x^4/4! +...) = ex-1-x
(1 + x + x^2/2! + x^3/3! +...)3 = e3x
The Attempt at a Solution
[(x^2/2! + x^3/3! + x^4/4! +...) =...
Homework Statement
let X1, X2,... Xn form a random sample of size n from the exponential distribution whose pdf if f(x|B) = Be-Bx for x>0 and B>0. Find an unbiased estimator of B.
Homework Equations
The Attempt at a Solution
nothing yet. i don't really know where to get started. a...
Homework Statement
find the Fourier transform of complex exponential multiplied to a unit step.
given: v(t)=exp(-i*wo*t)*u(t)
Homework Equations
∫(v(t)*exp(-i*w*t) dt) from -∞ to +∞
The Attempt at a Solution
∫([v(t)]*exp(-i*w*t) dt) from -∞ to +∞...
this is just something has been bugging me for the last few days. it seems like it has a very basic solution.
Muons decay randomly, but have a mean lifetime of about 2 us. If I plot the # of muons that decay vs. time (say the axis spans from 0 to 20 us), why is the plot exponential decay...
Homework Statement
If I have a matrix M, say
30 5
20 16
How do I calculate M^{1870} mod 101 using Euler's Theorem.
Homework Equations
I have so far worked out M ^{2} mod 101 to be
91 28
11 53
and thought I could use this as 2x935=1870
The Attempt at a Solution
I...
Homework Statement
Annoying problem, I recently got a graphing calculator (t1-89) and according to instructions i can calculate the i (interest) on it but each problem before i attempt on a graphing calculator i attempt to solve myself manually without a graphing calculator as a safeguard to...
Homework Statement
The question is located here http://i51.tinypic.com/nex2q1.jpg
Homework Equations
The value I have been given for a) is 5
The value I have been given for b) is 5PI/6
The Attempt at a Solution
Note that e^(a + bi) = e^a e^(bi) = e^a (cos b + i sin b).
(e^a is...
Homework Statement
Water is pumped into a tank. Volume V, is kept constant by continuos flow. The amount of salt S, depends on the amount of water that ahs been pumped in, call it X.
ds/dx = -S/V
Find the amount of water needed to eliminate 50% of the salt. Take v AS 10,000 gallons...
exponential equation (simple??)
x = Ae ^ kt
initally (at t = 0) x = 0.2 and when t = 2 then x = 1.5
a) Find A and k
b) Find x when t = 1.5
c) How long will it take x to decay to x = 0.01
Really struggling, any help would be greatly appreciated.
Homework Statement
\lim_{x \to \infty} \left(e^x-x \right )^{1/x}
Homework Equations
The Attempt at a Solution
Should I be equating this to f(x), taking the log of both sides, using L'Hopital's rule on the resulting indeterminate quotient? If I do that, then I end up with a perpetual...
I need to prove that ez1 x ez2 = e(z1 + z2)
using the power series ez = (SUM FROM n=0 to infinity) zn/n!
(For some reason the Sigma operator isn't working)
In the proof I have been given, it reads
(SUM from 0 to infinity) z1n/n! x (SUM from 0 to infinity)z2m/m!
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Homework Statement
Problem: find t in the following equationHomework Equations
64000e^{-1600t}+4000e^{-400t}=50000e^{-1000t}
The Attempt at a Solution
I know the answer: t=6.17\cdot\;10^{-4}s. But I'm struggling with how to get there. This is my attempt:
Factorizing down to...
Homework Statement
A radioactive substance has a half-life of 20 years. If 8 mg of the substance remains after
100 years, find how much of the substance was initially present.
Homework Equations
A=A0ekt
The Attempt at a Solution
I set the equation up so that 8=A0e100k and...
Homework Statement
In the 2 following problems they use the term in the brackets differently, in one case its a percentage and in the other case i have no idea where they get the number from, this is what i would like to find out
A cell loses 2% of its charge every day
C is total charge t...
Homework Statement
I have to evaluate P(t)=|<+,n|\exp{\frac{-iHt}{\hbar}}|+,n>|^2 where H=\hbar \omega_0 S_z + \hbar \omega a^+a+\hbar \lambda(a^+S_-+aS_+) and |+,n>=\left( \begin{array}{c}
1\\0 \end{array} \right)
Homework Equations
Eigenvalues of H are E_\pm =\hbar \omega (n...